Traders often price illiquid financial instruments whose price depends on volatility, such as illiquid options, based on a pricing model that uses, in part, a two-dimensional volatility surface (or grid). The volatility surface typically is calibrated based on liquid financial instruments, and more accurate pricing is obtained when the calibrated surface is smooth, so that sharp, isolated spikes in the surface do not cause distorted pricing outputs from the pricing model.
Presently, typically about fifty (50) liquid instruments are used to calibrate a volatility grid, but the grids typically include on the order of 20,000 points (or pixels). This presents an ill-posed inverse problem to calibrate the surface. Currently employed techniques to calibrate the surface are computationally intensive and complex, and still do not often produce smooth results.
For example, in one known technique for calibrating a volatility grid (“volgrid”), because different calibration instruments will depend, in general, on different areas (combinations of pixels) of the volgrid, non-overlapping partition areas covering sensitivity areas of the calibration instruments are defined. For reasonably selected instruments, it is possible to calibrate the volgrid by applying parallel shifts of the volgrid surface in those areas. Because the resulting volgrid will be discontinuous, it can be smoothed, and calibrated again. The iterations may be repeated until the process converges. In another known technique, volgrid elements are adjusted at each calibration step by solving a large-scale optimization problem, based on a large Jacobian size matrix (comprising sensitivities of instrument prices to changes of volgrid elements) and a smoothness penalty function. Both of these known techniques are computationally intensive, time consuming, and complex.